/**
* Calculates the greatest common divisor and Bezout's identity.
*
* Say you have 693 and 609. The GCD is 21. Bezout's identity states that there exist integers x and y such that
* 693*x + 609*y == 21. In point of fact, there are actually an infinite number of x and y combinations and which
* combination is returned is dependant upon which mode is in use. See
* {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bezout's identity - Wikipedia} for more information.
*
* Here's an example:
* <code>
* <?php
* include 'Math/BigInteger.php';
*
* $a = new Math_BigInteger(693);
* $b = new Math_BigInteger(609);
*
* extract($a->extendedGCD($b));
*
* echo $gcd->toString() . "\r\n"; // outputs 21
* echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21
* ?>
* </code>
*
* @param Math_BigInteger $n
* @return Math_BigInteger
* @access public
* @internal Calculates the GCD using the binary xGCD algorithim described in
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,
* the more traditional algorithim requires "relatively costly multiple-precision divisions".
*/
function extendedGCD($n)
{
switch (MATH_BIGINTEGER_MODE) {
case MATH_BIGINTEGER_MODE_GMP:
extract(gmp_gcdext($this->value, $n->value));
return array('gcd' => $this->_normalize(new Math_BigInteger($g)), 'x' => $this->_normalize(new Math_BigInteger($s)), 'y' => $this->_normalize(new Math_BigInteger($t)));
case MATH_BIGINTEGER_MODE_BCMATH:
// it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
// best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is,
// the basic extended euclidean algorithim is what we're using.
$u = $this->value;
$v = $n->value;
$a = '1';
$b = '0';
$c = '0';
$d = '1';
while (bccomp($v, '0', 0) != 0) {
$q = bcdiv($u, $v, 0);
$temp = $u;
$u = $v;
$v = bcsub($temp, bcmul($v, $q, 0), 0);
$temp = $a;
$a = $c;
$c = bcsub($temp, bcmul($a, $q, 0), 0);
$temp = $b;
$b = $d;
$d = bcsub($temp, bcmul($b, $q, 0), 0);
}
return array('gcd' => $this->_normalize(new Math_BigInteger($u)), 'x' => $this->_normalize(new Math_BigInteger($a)), 'y' => $this->_normalize(new Math_BigInteger($b)));
}
$y = $n->copy();
$x = $this->copy();
$g = new Math_BigInteger();
$g->value = array(1);
while (!($x->value[0] & 1 || $y->value[0] & 1)) {
$x->_rshift(1);
$y->_rshift(1);
$g->_lshift(1);
}
$u = $x->copy();
$v = $y->copy();
$a = new Math_BigInteger();
$b = new Math_BigInteger();
$c = new Math_BigInteger();
$d = new Math_BigInteger();
$a->value = $d->value = $g->value = array(1);
$b->value = $c->value = array();
while (!empty($u->value)) {
while (!($u->value[0] & 1)) {
$u->_rshift(1);
if (!empty($a->value) && $a->value[0] & 1 || !empty($b->value) && $b->value[0] & 1) {
$a = $a->add($y);
$b = $b->subtract($x);
}
$a->_rshift(1);
$b->_rshift(1);
}
while (!($v->value[0] & 1)) {
$v->_rshift(1);
if (!empty($d->value) && $d->value[0] & 1 || !empty($c->value) && $c->value[0] & 1) {
$c = $c->add($y);
$d = $d->subtract($x);
}
$c->_rshift(1);
$d->_rshift(1);
}
if ($u->compare($v) >= 0) {
$u = $u->subtract($v);
$a = $a->subtract($c);
$b = $b->subtract($d);
} else {
$v = $v->subtract($u);
$c = $c->subtract($a);
$d = $d->subtract($b);
}
}
return array('gcd' => $this->_normalize($g->multiply($v)), 'x' => $this->_normalize($c), 'y' => $this->_normalize($d));
}
/**
* Create public / private key pair
*
* Returns an array with the following three elements:
* - 'privatekey': The private key.
* - 'publickey': The public key.
* - 'partialkey': A partially computed key (if the execution time exceeded $timeout).
* Will need to be passed back to Crypt_RSA::createKey() as the third parameter for further processing.
*
* @access public
* @param int $bits
* @param int $timeout
* @param Math_BigInteger $p
*/
function createKey($bits = 1024, $timeout = false, $partial = array())
{
if (!defined('CRYPT_RSA_EXPONENT')) {
// http://en.wikipedia.org/wiki/65537_%28number%29
define('CRYPT_RSA_EXPONENT', '65537');
}
// per <http://cseweb.ucsd.edu/~hovav/dist/survey.pdf#page=5>, this number ought not result in primes smaller
// than 256 bits. as a consequence if the key you're trying to create is 1024 bits and you've set CRYPT_RSA_SMALLEST_PRIME
// to 384 bits then you're going to get a 384 bit prime and a 640 bit prime (384 + 1024 % 384). at least if
// CRYPT_RSA_MODE is set to CRYPT_RSA_MODE_INTERNAL. if CRYPT_RSA_MODE is set to CRYPT_RSA_MODE_OPENSSL then
// CRYPT_RSA_SMALLEST_PRIME is ignored (ie. multi-prime RSA support is more intended as a way to speed up RSA key
// generation when there's a chance neither gmp nor OpenSSL are installed)
if (!defined('CRYPT_RSA_SMALLEST_PRIME')) {
define('CRYPT_RSA_SMALLEST_PRIME', 4096);
}
// OpenSSL uses 65537 as the exponent and requires RSA keys be 384 bits minimum
if (CRYPT_RSA_MODE == CRYPT_RSA_MODE_OPENSSL && $bits >= 384 && CRYPT_RSA_EXPONENT == 65537) {
$config = array();
if (isset($this->configFile)) {
$config['config'] = $this->configFile;
}
$rsa = openssl_pkey_new(array('private_key_bits' => $bits) + $config);
openssl_pkey_export($rsa, $privatekey, null, $config);
$publickey = openssl_pkey_get_details($rsa);
$publickey = $publickey['key'];
$privatekey = call_user_func_array(array($this, '_convertPrivateKey'), array_values($this->_parseKey($privatekey, CRYPT_RSA_PRIVATE_FORMAT_PKCS1)));
$publickey = call_user_func_array(array($this, '_convertPublicKey'), array_values($this->_parseKey($publickey, CRYPT_RSA_PUBLIC_FORMAT_PKCS1)));
// clear the buffer of error strings stemming from a minimalistic openssl.cnf
while (openssl_error_string() !== false) {
}
return array('privatekey' => $privatekey, 'publickey' => $publickey, 'partialkey' => false);
}
static $e;
if (!isset($e)) {
$e = new Math_BigInteger(CRYPT_RSA_EXPONENT);
}
extract($this->_generateMinMax($bits));
$absoluteMin = $min;
$temp = $bits >> 1;
// divide by two to see how many bits P and Q would be
if ($temp > CRYPT_RSA_SMALLEST_PRIME) {
$num_primes = floor($bits / CRYPT_RSA_SMALLEST_PRIME);
$temp = CRYPT_RSA_SMALLEST_PRIME;
} else {
$num_primes = 2;
}
extract($this->_generateMinMax($temp + $bits % $temp));
$finalMax = $max;
extract($this->_generateMinMax($temp));
$generator = new Math_BigInteger();
$n = $this->one->copy();
if (!empty($partial)) {
extract(unserialize($partial));
} else {
$exponents = $coefficients = $primes = array();
$lcm = array('top' => $this->one->copy(), 'bottom' => false);
}
$start = time();
$i0 = count($primes) + 1;
do {
for ($i = $i0; $i <= $num_primes; $i++) {
if ($timeout !== false) {
$timeout -= time() - $start;
$start = time();
if ($timeout <= 0) {
return array('privatekey' => '', 'publickey' => '', 'partialkey' => serialize(array('primes' => $primes, 'coefficients' => $coefficients, 'lcm' => $lcm, 'exponents' => $exponents)));
}
}
if ($i == $num_primes) {
list($min, $temp) = $absoluteMin->divide($n);
if (!$temp->equals($this->zero)) {
$min = $min->add($this->one);
// ie. ceil()
}
$primes[$i] = $generator->randomPrime($min, $finalMax, $timeout);
} else {
$primes[$i] = $generator->randomPrime($min, $max, $timeout);
}
if ($primes[$i] === false) {
// if we've reached the timeout
if (count($primes) > 1) {
$partialkey = '';
} else {
array_pop($primes);
$partialkey = serialize(array('primes' => $primes, 'coefficients' => $coefficients, 'lcm' => $lcm, 'exponents' => $exponents));
}
return array('privatekey' => '', 'publickey' => '', 'partialkey' => $partialkey);
}
// the first coefficient is calculated differently from the rest
// ie. instead of being $primes[1]->modInverse($primes[2]), it's $primes[2]->modInverse($primes[1])
if ($i > 2) {
$coefficients[$i] = $n->modInverse($primes[$i]);
}
$n = $n->multiply($primes[$i]);
$temp = $primes[$i]->subtract($this->one);
// textbook RSA implementations use Euler's totient function instead of the least common multiple.
// see http://en.wikipedia.org/wiki/Euler%27s_totient_function
$lcm['top'] = $lcm['top']->multiply($temp);
$lcm['bottom'] = $lcm['bottom'] === false ? $temp : $lcm['bottom']->gcd($temp);
$exponents[$i] = $e->modInverse($temp);
}
list($temp) = $lcm['top']->divide($lcm['bottom']);
$gcd = $temp->gcd($e);
$i0 = 1;
} while (!$gcd->equals($this->one));
$d = $e->modInverse($temp);
$coefficients[2] = $primes[2]->modInverse($primes[1]);
// from <http://tools.ietf.org/html/rfc3447#appendix-A.1.2>:
// RSAPrivateKey ::= SEQUENCE {
// version Version,
// modulus INTEGER, -- n
// publicExponent INTEGER, -- e
// privateExponent INTEGER, -- d
// prime1 INTEGER, -- p
// prime2 INTEGER, -- q
// exponent1 INTEGER, -- d mod (p-1)
// exponent2 INTEGER, -- d mod (q-1)
// coefficient INTEGER, -- (inverse of q) mod p
// otherPrimeInfos OtherPrimeInfos OPTIONAL
// }
return array('privatekey' => $this->_convertPrivateKey($n, $e, $d, $primes, $exponents, $coefficients), 'publickey' => $this->_convertPublicKey($n, $e), 'partialkey' => false);
}
